Statistical methods for analysis of time-to-event (survival) and longitudinal data are fundamental tools for research in cancer and other diseases, and continued progress on new methods is essential. New methods are proposed that unify, extend, or enhance existing procedures. Popular methods for analysis of censored survival data are non- or semiparametric. New methods are proposed that rest on the contention that, if one is willing to make mild "smoothness" assumptions that move the problem away slightly from traditional approaches, improvement may be possible; moreover, computational advantages may emerge. The procedures exploit a method favored in econometrics and offer the potential for a unified approach to many such problems that have traditionally been considered distinct. The ubiquitous application of the Cox proportional hazards model, almost by default, highlights the need for continued development of methods to detect departures from the proportional hazards assumption. A new method that also suggests a general approach to testing covariates effects is proposed. There has been considerable interest in longitudinal and survival analysis on the implications of deviation from parametric assumptions in models involving unobservable random effects or other latent variables, such as mixed effects, measurement error, and joint mixed effects-survival models. Recent has revealed surprising empirical evidence of robustness of popular parametric models to misspecification; however, no systematic understanding has emerged. A formal approach to studying model robustness, which suggests practical diagnostic tools, is proposed. Alternatively, new methods for longitudinal data analysis that circumvent this issue altogether will be developed.